Dr. Potter provides vaccinations against polio and measles. Each polio vaccination consists of $4$ doses, and each measles vaccination consists of $2$ doses. Last year, Dr. Potter gave a total of $60$ vaccinations that consisted of a total of $184$ doses. How many polio vaccinations and how many measles vaccinations did Dr. Potter give last year? Dr. Potter gave
Solution: Let $x$ represent the number of polio vaccinations Dr. Potter gave and let $y$ represent the number of measles vaccinations she gave. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that each polio vaccine consists of $\textit{4}$ doses, each measles vaccine consists of $\textit{2}$ doses, and Dr. Potter gave a total of $\textit{184}$ doses of vaccination last year. How can we model this sentence algebraically? The total number of doses of the polio vaccines received by children can be modeled by $4x$, and the total number of doses of the measles vaccines received by children can be modeled by $2y$. Since these together add up to $184$, we get the following equation: $4 x+2 y=184$ We are also given that Dr. Potter gave a total of $\textit{60}$ vaccinations last year. This can be expressed as: $x+y=60$ Let's rewrite this equation so that it's solved for $y$ : $y = 60-x$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ y={60-x}$ into the first equation: $\begin{aligned}4x+ 2 y &= 184\\\\ 4x+2\cdot ({60-x})&=184\\\\ 4x+120-2x&=184\\\\ 2x &=64\\\\ x&=32\end{aligned}$ Now we can substitute $x = 32$ into $y=60-x$ and find that $y=28$. Recall that $x$ denotes the number of polio vaccinations and $y$ denotes the number of measles vaccinations. Therefore, Dr. Potter gave $\textit{32}$ polio vaccinations and $\textit{28}$ measles vaccinations.